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INEQUALITIES FOR THE ANGULAR DERIVATIVES OF CERTAIN CLASSES OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISC

  • Ornek, Bulent Nafi (Department of Computer Engineering, Amasya University)
  • Received : 2013.10.08
  • Published : 2016.03.31

Abstract

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$ holomorphic in the unit disc and $\|\frac{f(z)}{{\lambda}f(z)+(1-{\lambda})z}-{\alpha}\|$ < ${\alpha}$ for ${\mid}z{\mid}$ < 1, where $\frac{1}{2}$ < ${\alpha}$ ${\leq}{\frac{1}{1+{\lambda}}}$, $0{\leq}{\lambda}$ < 1. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+{\cdots}$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $c_{p+1}$, $c_{p+2}$ and zeros of f(z) - z. We obtain a sharp lower bound of ${\mid}f^{\prime}(b){\mid}$ at the point b, where ${\mid}b{\mid}=1$.

Keywords

Schwarz lemma on the boundary;holomorphic function;angular derivative;Julia-Wolff-Lemma

References

  1. T. Aliyev Azeroglu and B. N. Ornek, A refined Schwarz inequality on the boundary, Complex Var. Elliptic Equ. 58 (2013), no. 4, 571-577. https://doi.org/10.1080/17476933.2012.718338
  2. V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. (N. Y.) 122 (2004), no. 6, 3623-3629. https://doi.org/10.1023/B:JOTH.0000035237.43977.39
  3. V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. (N. Y.) 207 (2015), no. 6, 825-831. https://doi.org/10.1007/s10958-015-2406-5
  4. G. M. Golusin, Geometric Theory of Functions of Complex Variable, 2nd edn., Moscow 1966.
  5. M. Jeong, The Schwarz lemma and its application at a boundary point, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 3, 219-227.
  6. B. N. Ornek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), no. 6, 2053-2059. https://doi.org/10.4134/BKMS.2013.50.6.2053
  7. R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0
  8. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
  9. H. Unkelbach, Uber die Randverzerrung bei konformer Abbildung, Math. Z. 43 (1938), no. 1, 739-742. https://doi.org/10.1007/BF01181115